The modal verbs are can, could, may, might, must, will, would, should, ought to. They are known as modal auxiliary verbs because they "help" another verb.

I can swim.
Do you think I should go?

Form

1. There is no -s in the third person singular.

She can ski.
He must be tired.
It might rain.

2. There is no do/does/doesn't in the question or negative.

What should I do?
Can I help you?
You mustn't steal!
He can't dance.

3. Modal auxiliary verbs are followed by the infinitive without to. The exception is ought to.

You must go.
I'll help you.
You ought to see a doctor.

4. They have no infinitives and no -ing forms. Other expressions are used instead.

I'd love to be able to ski.
I hate having to get up on cold, winter mornings.

5. They don't usually have past forms. Instead, we can use them with perfect infinitives.

You should have told me that you can't swim.
You might have drowned!

or we use other expressions:

I had to work hard in school.

Note: Could is used with a past meaning to talk about a general ability,

I could swim when I was six. (general ability)

To talk about ability on one specific occasion, we use was able to / managed to.

The prisoner was able to / managed to escape by climbing on the roof of the prison (NOT could escape).

Use

1. Modal verbs express our attitudes, opinions, and judgments of events. Compare:

"Who's that knocking on the door?"
"It's John." (This is a fact.)

"Who's that knocking on the door?"
"It could / may / might / must / should / can't /will be John. (These all express our attitude or opinion.)

2. Each modal verb has at least two meanings. One use of all of them is to express possibility or probability.

I must post this letter! (obligation)
You must be tired! (deduction, probability)
Could you help me? (polite request)
We could go to Spain for our holiday. (possibility)
You may go home now (permission)
"Where's Anna?" "I'm not sure. She may be at work." (possibility)

4.1. Modal verbs of obligation and permission.

Have to, have got to.

Negative and positive forms

I have to work hard.
You have to work hard.
He has to work hard.
She doesn't have to work hard.
We have to work hard.
You don't have to work hard.
They don't have to work hard.

Question

Do you have to work hard?
Do I have to work hard?

Use

Have to is not a modal verb.

1. Have to expresses strong obligation. It expresses a general obligation based on a law or rule, based on the authority of another person. It's impersonal.

Children have to go to school until they are 16. (a law)
Mum says you have to clean your room before you go out. (mother's order)

2. Have got to is common in British English but it's more informal than have to.

I've got to go no. See you!
Don't go to bed late. We've got to get up early tomorrow.
"Go and tidy your room." "Have I got to?" "Yes, you have!"

3. Have to expresses a general repeated obligation.

I always have to tell my parents where I'm going.

Have got to expresses an obligation on one particular occasion.

I've got to get up early tomorrow to catch a train.

Modal verbs of obligation and permission.

Can and be allowed to.

Affirmative and negative forms

I can park here.
You can't park here.
He isn't allowed to park here.
She is allowed to park here.
We are allowed to park here.
You aren't allowed to park here.
They are not allowed to park here.

Question

Can you park here?
Are you allowed to park here?
Is he allowed to park here?
Am I allowed to park here?
Can she park here?

Use

Can is a modal verb.

Can and be allowed to express permission. Can is more informal and usually spoken.

You can borrow my bike, but you can't have the car. I need it.
They can't come in here with those muddy shoes!
You're allowed to get married when you're 18.
Are we allowed to use a dictionary for this test?
He isn't allowed to park here.